Calculus Examples

$f (x) = 6 {(4 x + 1)}^{4} {(3 x - 5)}^{2}$

Step 1

Rewrite ${(3 x - 5)}^{2}$ as $(3 x - 5) (3 x - 5)$ .

$\frac{d}{d x} [6 {(4 x + 1)}^{4} ((3 x - 5) (3 x - 5))]$

Step 2

Expand $(3 x - 5) (3 x - 5)$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$\frac{d}{d x} [6 {(4 x + 1)}^{4} (3 x (3 x - 5) - 5 (3 x - 5))]$

Step 2.2

Apply the distributive property.

$\frac{d}{d x} [6 {(4 x + 1)}^{4} (3 x (3 x) + 3 x \cdot - 5 - 5 (3 x - 5))]$

Step 2.3

Apply the distributive property.

$\frac{d}{d x} [6 {(4 x + 1)}^{4} (3 x (3 x) + 3 x \cdot - 5 - 5 (3 x) - 5 \cdot - 5)]$

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Rewrite using the commutative property of multiplication.

$\frac{d}{d x} [6 {(4 x + 1)}^{4} (3 \cdot 3 x \cdot x + 3 x \cdot - 5 - 5 (3 x) - 5 \cdot - 5)]$

Step 3.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 3.1.2.1

Move $x$ .

$\frac{d}{d x} [6 {(4 x + 1)}^{4} (3 \cdot 3 (x \cdot x) + 3 x \cdot - 5 - 5 (3 x) - 5 \cdot - 5)]$

Step 3.1.2.2

Multiply $x$ by $x$ .

$\frac{d}{d x} [6 {(4 x + 1)}^{4} (3 \cdot 3 x^{2} + 3 x \cdot - 5 - 5 (3 x) - 5 \cdot - 5)]$

Step 3.1.3

Multiply $3$ by $3$ .

$\frac{d}{d x} [6 {(4 x + 1)}^{4} (9 x^{2} + 3 x \cdot - 5 - 5 (3 x) - 5 \cdot - 5)]$

Step 3.1.4

Multiply $- 5$ by $3$ .

$\frac{d}{d x} [6 {(4 x + 1)}^{4} (9 x^{2} - 15 x - 5 (3 x) - 5 \cdot - 5)]$

Step 3.1.5

Multiply $3$ by $- 5$ .

$\frac{d}{d x} [6 {(4 x + 1)}^{4} (9 x^{2} - 15 x - 15 x - 5 \cdot - 5)]$

Step 3.1.6

Multiply $- 5$ by $- 5$ .

$\frac{d}{d x} [6 {(4 x + 1)}^{4} (9 x^{2} - 15 x - 15 x + 25)]$

Step 3.2

Subtract $15 x$ from $- 15 x$ .

$\frac{d}{d x} [6 {(4 x + 1)}^{4} (9 x^{2} - 30 x + 25)]$

Step 4

Since $6$ is constant with respect to $x$ , the derivative of $6 {(4 x + 1)}^{4} (9 x^{2} - 30 x + 25)$ with respect to $x$ is $6 \frac{d}{d x} [{(4 x + 1)}^{4} (9 x^{2} - 30 x + 25)]$ .

$6 \frac{d}{d x} [{(4 x + 1)}^{4} (9 x^{2} - 30 x + 25)]$

Step 5

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = {(4 x + 1)}^{4}$ and $g (x) = 9 x^{2} - 30 x + 25$ .

$6 ({(4 x + 1)}^{4} \frac{d}{d x} [9 x^{2} - 30 x + 25] + (9 x^{2} - 30 x + 25) \frac{d}{d x} [{(4 x + 1)}^{4}])$

Step 6

Differentiate.

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Step 6.1

By the Sum Rule, the derivative of $9 x^{2} - 30 x + 25$ with respect to $x$ is $\frac{d}{d x} [9 x^{2}] + \frac{d}{d x} [- 30 x] + \frac{d}{d x} [25]$ .

$6 ({(4 x + 1)}^{4} (\frac{d}{d x} [9 x^{2}] + \frac{d}{d x} [- 30 x] + \frac{d}{d x} [25]) + (9 x^{2} - 30 x + 25) \frac{d}{d x} [{(4 x + 1)}^{4}])$

Step 6.2

Since $9$ is constant with respect to $x$ , the derivative of $9 x^{2}$ with respect to $x$ is $9 \frac{d}{d x} [x^{2}]$ .

$6 ({(4 x + 1)}^{4} (9 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 30 x] + \frac{d}{d x} [25]) + (9 x^{2} - 30 x + 25) \frac{d}{d x} [{(4 x + 1)}^{4}])$

Step 6.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$6 ({(4 x + 1)}^{4} (9 (2 x) + \frac{d}{d x} [- 30 x] + \frac{d}{d x} [25]) + (9 x^{2} - 30 x + 25) \frac{d}{d x} [{(4 x + 1)}^{4}])$

Step 6.4

Multiply $2$ by $9$ .

$6 ({(4 x + 1)}^{4} (18 x + \frac{d}{d x} [- 30 x] + \frac{d}{d x} [25]) + (9 x^{2} - 30 x + 25) \frac{d}{d x} [{(4 x + 1)}^{4}])$

Step 6.5

Since $- 30$ is constant with respect to $x$ , the derivative of $- 30 x$ with respect to $x$ is $- 30 \frac{d}{d x} [x]$ .

$6 ({(4 x + 1)}^{4} (18 x - 30 \frac{d}{d x} [x] + \frac{d}{d x} [25]) + (9 x^{2} - 30 x + 25) \frac{d}{d x} [{(4 x + 1)}^{4}])$

Step 6.6

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$6 ({(4 x + 1)}^{4} (18 x - 30 \cdot 1 + \frac{d}{d x} [25]) + (9 x^{2} - 30 x + 25) \frac{d}{d x} [{(4 x + 1)}^{4}])$

Step 6.7

Multiply $- 30$ by $1$ .

$6 ({(4 x + 1)}^{4} (18 x - 30 + \frac{d}{d x} [25]) + (9 x^{2} - 30 x + 25) \frac{d}{d x} [{(4 x + 1)}^{4}])$

Step 6.8

Since $25$ is constant with respect to $x$ , the derivative of $25$ with respect to $x$ is $0$ .

$6 ({(4 x + 1)}^{4} (18 x - 30 + 0) + (9 x^{2} - 30 x + 25) \frac{d}{d x} [{(4 x + 1)}^{4}])$

Step 6.9

Add $18 x - 30$ and $0$ .

$6 ({(4 x + 1)}^{4} (18 x - 30) + (9 x^{2} - 30 x + 25) \frac{d}{d x} [{(4 x + 1)}^{4}])$

Step 7

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{4}$ and $g (x) = 4 x + 1$ .

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Step 7.1

To apply the Chain Rule, set $u$ as $4 x + 1$ .

$6 ({(4 x + 1)}^{4} (18 x - 30) + (9 x^{2} - 30 x + 25) (\frac{d}{d u} [u^{4}] \frac{d}{d x} [4 x + 1]))$

Step 7.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 4$ .

$6 ({(4 x + 1)}^{4} (18 x - 30) + (9 x^{2} - 30 x + 25) (4 u^{3} \frac{d}{d x} [4 x + 1]))$

Step 7.3

Replace all occurrences of $u$ with $4 x + 1$ .

$6 ({(4 x + 1)}^{4} (18 x - 30) + (9 x^{2} - 30 x + 25) (4 {(4 x + 1)}^{3} \frac{d}{d x} [4 x + 1]))$

Step 8

Differentiate.

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Step 8.1

Move $4$ to the left of $9 x^{2} - 30 x + 25$ .

$6 ({(4 x + 1)}^{4} (18 x - 30) + 4 \cdot (9 x^{2} - 30 x + 25) {(4 x + 1)}^{3} \frac{d}{d x} [4 x + 1])$

Step 8.2

By the Sum Rule, the derivative of $4 x + 1$ with respect to $x$ is $\frac{d}{d x} [4 x] + \frac{d}{d x} [1]$ .

$6 ({(4 x + 1)}^{4} (18 x - 30) + 4 (9 x^{2} - 30 x + 25) {(4 x + 1)}^{3} (\frac{d}{d x} [4 x] + \frac{d}{d x} [1]))$

Step 8.3

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

$6 ({(4 x + 1)}^{4} (18 x - 30) + 4 (9 x^{2} - 30 x + 25) {(4 x + 1)}^{3} (4 \frac{d}{d x} [x] + \frac{d}{d x} [1]))$

Step 8.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$6 ({(4 x + 1)}^{4} (18 x - 30) + 4 (9 x^{2} - 30 x + 25) {(4 x + 1)}^{3} (4 \cdot 1 + \frac{d}{d x} [1]))$

Step 8.5

Multiply $4$ by $1$ .

$6 ({(4 x + 1)}^{4} (18 x - 30) + 4 (9 x^{2} - 30 x + 25) {(4 x + 1)}^{3} (4 + \frac{d}{d x} [1]))$

Step 8.6

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$6 ({(4 x + 1)}^{4} (18 x - 30) + 4 (9 x^{2} - 30 x + 25) {(4 x + 1)}^{3} (4 + 0))$

Step 8.7

Simplify the expression.

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Step 8.7.1

Add $4$ and $0$ .

$6 ({(4 x + 1)}^{4} (18 x - 30) + 4 (9 x^{2} - 30 x + 25) {(4 x + 1)}^{3} \cdot 4)$

Step 8.7.2

Multiply $4$ by $4$ .

$6 ({(4 x + 1)}^{4} (18 x - 30) + 16 (9 x^{2} - 30 x + 25) {(4 x + 1)}^{3})$

Step 9

Simplify.

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Step 9.1

Apply the distributive property.

$6 ({(4 x + 1)}^{4} (18 x - 30) + (16 (9 x^{2}) + 16 (- 30 x) + 16 \cdot 25) {(4 x + 1)}^{3})$

Step 9.2

Apply the distributive property.

$6 ({(4 x + 1)}^{4} (18 x - 30)) + 6 ((16 (9 x^{2}) + 16 (- 30 x) + 16 \cdot 25) {(4 x + 1)}^{3})$

Step 9.3

Multiply $9$ by $16$ .

$6 {(4 x + 1)}^{4} (18 x - 30) + 6 ((144 x^{2} + 16 (- 30 x) + 16 \cdot 25) {(4 x + 1)}^{3})$

Step 9.4

Multiply $- 30$ by $16$ .

$6 {(4 x + 1)}^{4} (18 x - 30) + 6 ((144 x^{2} - 480 x + 16 \cdot 25) {(4 x + 1)}^{3})$

Step 9.5

Multiply $16$ by $25$ .

$6 {(4 x + 1)}^{4} (18 x - 30) + 6 ((144 x^{2} - 480 x + 400) {(4 x + 1)}^{3})$

Step 9.6

Factor $6 {(4 x + 1)}^{3}$ out of $6 {(4 x + 1)}^{4} (18 x - 30) + 6 (144 x^{2} - 480 x + 400) {(4 x + 1)}^{3}$ .

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Step 9.6.1

Factor $6 {(4 x + 1)}^{3}$ out of $6 {(4 x + 1)}^{4} (18 x - 30)$ .

$6 {(4 x + 1)}^{3} ((4 x + 1) (18 x - 30)) + 6 (144 x^{2} - 480 x + 400) {(4 x + 1)}^{3}$

Step 9.6.2

Factor $6 {(4 x + 1)}^{3}$ out of $6 (144 x^{2} - 480 x + 400) {(4 x + 1)}^{3}$ .

$6 {(4 x + 1)}^{3} ((4 x + 1) (18 x - 30)) + 6 {(4 x + 1)}^{3} (144 x^{2} - 480 x + 400)$

Step 9.6.3

Factor $6 {(4 x + 1)}^{3}$ out of $6 {(4 x + 1)}^{3} ((4 x + 1) (18 x - 30)) + 6 {(4 x + 1)}^{3} (144 x^{2} - 480 x + 400)$ .

$6 {(4 x + 1)}^{3} ((4 x + 1) (18 x - 30) + 144 x^{2} - 480 x + 400)$